Problem: Solve the exponential equation for $x$. 27 9 x + 4 ⋅ 3 x − 8 = 3 x + 9 27\^{9x+4}\cdot 3\^{ x-8}=3\^{ x+9} $x=$
Explanation: The strategy Let's write $27$ in base $3$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $3$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 27 9 x + 4 ⋅ 3 x − 8 = ( 3 3 ) 9 x + 4 ⋅ 3 x − 8 = 3 27 x + 12 ⋅ 3 x − 8 = 3 27 x + 12 + ( x − 8 ) = 3 28 x + 4 ( 27 = 3 3 ) ( ( a n ) m = a n ⋅ m ) ( a n ⋅ a m = a n + m ) \begin{aligned} 27\^{9x+4}\cdot 3\^{ x-8}&=(3^3)\^{9x+4}\cdot 3\^{ x-8}&&&&(27=3^3)\\\\ &=3\^{C{27x+12}}\cdot 3\^{ {x-8}}&&&&((a^n)^m=a^{n\cdot m}) \\\\ &=3\^{ C{27x+12} \ + \ ({x-8}) }&&&&(a^n\cdot a^m=a^{n + \normalsize m})\\\\ &=3\^{ 28x+4} \end{aligned} Solving the linear equation We obtain the following equation. 3 28 x + 4 = 3 x + 9 3\^{ 28x+4}=3\^{ x+9} Now we can equate the exponents and solve for $x$. $\begin{aligned} 28x+4 &=x+9\\\\ x &= \dfrac{5}{27}\end{aligned}$ The answer The answer is $x=\dfrac{5}{27}$. You can check this answer by substituting $\it{x=\dfrac{5}{27}}$ in the original equation and evaluating both sides.